The exponent that came free
A recent paper reports that a warning lead-time grows super-linearly with a timescale — a power law with exponent above 1, read as a real dynamical scaling law. It’s a clean, quotable summary. It is also, in a specific sense, a fact about a definition rather than about the dynamics.
To be clear up front about scope: this re-analysis uses the paper’s own published tables, and it does not dispute the paper’s primary contribution — that a geometric defect rises before grokking. It examines one secondary claim: the characterization of the lead time Δt = c·tgrokα as super-linear (reported α = 1.180 for SCAN, 1.132 for Dyck). The exponent and R² reproduce exactly by OLS on the reported data, which confirms the method matches.
1. The exponent is largely mechanical
The lead time is defined as Δt = tgrok − tonset. Across the reported runs, onset is a small fraction of tgrok (lead fractions are 90%+ for most runs), so Δt ≈ tgrok, and regressing Δt on tgrok is close to regressing tgrok on itself.
As a check, replace the true onset times with a constant — no dynamics at all — and refit. That already yields α = 1.47 (SCAN), 1.07 (Dyck). So α > 1 is the baseline you get from the definition alone, before any signal behavior enters. In fact the measured SCAN α sits below its mechanical baseline, because the real onset times grow with tgrok and pull α down toward 1.
2. The Dyck exponent hinges on one point
The Dyck values (α = 1.132, R² = 0.908) reproduce exactly only when all 15 runs are included — among them the single lowest-lead-fraction, smallest-scale run (lead fraction 16.7%). Remove that one point and the fit gives α = 0.967 (below 1) with a higher R² (0.978). By fit quality you would exclude it, and the exponent then falls below unity — reversing the super-linear reading.
| Fit | Reported / full | Under the check |
|---|---|---|
| SCAN exponent α | 1.180 | 1.47 from constant onset |
| Dyck exponent α | 1.132 (R²=0.908) | 0.967, R²=0.978 (drop 1 pt) |
| SCAN onset vs tgrok | tonset ~ tgrok0.276, CI [0.11, 0.38] — real sub-linear | |
| Dyck onset vs tgrok | tonset ~ tgrok0.762, CI [0.26, 1.43] — no robust scaling | |
3. The defensible object is onset-vs-timescale
Model the quantity the power law obscures — the onset time itself — against tgrok directly, and a real, task-specific statement appears. For SCAN, onset time grows sub-linearly (exponent 0.276, CI excludes 1): slower-grokking runs get proportionally earlier warning. For Dyck, the CI includes 1 — no robust scaling, consistent with a roughly constant lead fraction. So SCAN supports a genuine sub-linear onset-scaling; Dyck does not support any scaling claim once the right variable is analyzed.
Source: the paper’s published Tables (arXiv:2602.16967), lead times per run.
OLS on log(Δt) vs log(tgrok) reproduces α and R² exactly.
Constant-onset null → α=1.47 (SCAN), 1.07 (Dyck). Dyck all-15 α=1.132/R²=0.908 → drop lowest-lead run → α=0.967/R²=0.978.
Onset vs t_grok: SCAN 0.276 [0.11,0.38]; Dyck 0.762 [0.26,1.43].
Test fit fragility yourself:
evalgate loo points.txt --power-law --threshold 1.0